User blog:Granpa/Elementary mathematics
Positive numbers Addition :See 5+0\quad=\quad0+5\quad=\quad5 \\ 5+1\quad=\quad1+5\quad=\quad6 \\ 5+2\quad=\quad2+5\quad=\quad7 \\ 5+3\quad=\quad3+5\quad=\quad8 \\ 5+4\quad=\quad4+5\quad=\quad9 \\ 5+5\quad=\quad5+5\quad=\quad10 ¹ 27 + 59 ———— 86 Multiplication :See 5 \cdot 0 \quad = \quad 0 \cdot 5 \quad = \quad 0 \\ 5 \cdot 1 \quad = \quad 1 \cdot 5 \quad = \quad 5 \\ 5 \cdot 2 \quad = \quad 2 \cdot 5 \quad = \quad 5 + 5 \\ 5 \cdot 3 \quad = \quad 3 \cdot 5 \quad = \quad 5 + 5 + 5 \\ 5 \cdot 4 \quad = \quad 4 \cdot 5 \quad = \quad 5 + 5 + 5 + 5 | | |} 2 4 × 3 7 ------ 1 6 8 + 7 2 ---------- = 8 8 8 Powers :See 5^0 = 1 \\ 5^1 = 5 \\ 5^2 = 5 \cdot 5 \\ 5^3 = 5 \cdot 5 \cdot 5 \\ 5^4 = 5 \cdot 5 \cdot 5 \cdot 5 \\ 5^5 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\\ 5^2 \cdot 5^3 = (5 \cdot 5) \cdot (5 \cdot 5 \cdot 5) = 5^5 = 5^{(2+3)} \\ 5^2 ("five squared") is the area of a square that is 5 units wide. 5^3 ("five cubed") is the volume of a cube that is 5 units wide. As you can see in the following table, raising numbers to powers results in very large numbers. You will not need to memorize this table. It will occasionally be useful to know the squares of numbers but you can easily work that out in your head by multiplying the number times itself. From Wikipedia:Exponentiation: Negative numbers Subtraction Subtraction is the opposite of addition. : 5-3=2\quad because \quad2+3=5 But then what is : 3-5=? What number when added to 5 results in 3? The answer is that there is no answer. So we have to create one. We therefore create what are called negative numbers From Wikipedia:Negative number Addition with negative numbers Adding a negative number is the same as subtracting a positive number. :  5 − 3 2}}. Addition of two negative numbers is very similar to addition of two positive numbers. For example, :   −8}}. Subtraction with negative numbers Subtracting a negative number is the same as adding a positive number. :   5 + 3     8}} and :   (−5) + 8     3}}. Multiplication with negative numbers When multiplying numbers, the sign of the product is determined by the following rules: If two numbers have the same sign, the result is always positive ::   6}} :and ::   6}}. If the two numbers have different signs, the result is always negative ::   −6}} :and ::   −6}} Powers with negative numbers A negative number raised to an even number is positive :: -2^2=-2 \cdot -2 = 4 :and a negative number raised to an odd number is negative :: -2^3=-2 \cdot -2 \cdot -2 = -8 A number raised to a negative number is a fraction :: 2^{-1} = \frac{1}{2} :and :: 2^{-2}= \frac{1}{2 \cdot 2} = \frac{1}{4} Therefore : 5^{3} \cdot 5^{-3} = \frac{5 \cdot 5 \cdot 5}{5 \cdot 5 \cdot 5} = 1 = 5^0 = 5^{(3-3)} Fractions Division :See Division is the opposite of multiplication. :: \frac{6}{3} = 2 \quad because \quad 2 \cdot 3 = 6 :and :: \frac{2}{5} \cdot \frac{3}{4}=\frac{2 \cdot 3}{5 \cdot 4} = \frac{6}{20} = \frac{3}{10} \cdot \frac{2}{2}=\frac{3}{10} \cdot 1 = \frac{3}{10} :and :: \frac{500}{4}=125 :this can be worked out by hand like this 125 (Explanations) 4)500 4 ( 4 × 1 = 4) 10 ( 5 - 4 = 1) 8 ( 4 × 2 = 8) 20 (10 - 8 = 2) 20 ( 4 × 5 = 20) 0 (20 - 20 = 0) But then what is :: \frac{5}{4}=? The answer is that there is no answer. So we have to create one. So we create what are called decimal numbers. From Wikipedia:Long division: An example is shown below, representing the division of 5 by 4, with a result of 1.25 ("one point two five"). 1.25 (Explanations) 4)5.00 4 ( 4 × 1 = 4) 1.0 ( 5 - 4 = 1) 8 ( 4 × 2 = 8) 20 (10 - 8 = 2) 20 ( 4 × 5 = 20) 0 (20 - 20 = 0) The "." is called the . 31.75 4)127.00 12 (12 ÷ 4 = 3) 07 (0 remainder, bring down next figure) 4 (7 ÷ 4 = 1 r 3 ) 3.0 (0 is added in order to make 3 divisible by 4) 2.8 (7 × 4 = 28) 20 (an additional zero is brought down) 20 (5 × 4 = 20) 0 Some decimal numbers never end 0.333333333333333333333... 3)1.000000000000000000000... 9 1.0 9 10 9 10 9 10 9 10 The "..." at the end means that it just keeps repeating forever. This can also be indicated with an overline. The sign rules for division are the same as for multiplication. For example, If dividend and divisor have the same sign, the result is always positive. :: \phantom{-}\frac{8}{2}=\phantom{-}4 :and :: \frac{-8}{-2}=\phantom{-}4 If dividend and divisor have different signs, the result is always negative. :: \frac{\phantom{-}8}{-2}=-4 :and :: \frac{-8}{\phantom{-}2}=-4 Further reading *Introductory mathematics References Category:Blog posts